Properties

Label 450.6.a.q
Level $450$
Weight $6$
Character orbit 450.a
Self dual yes
Analytic conductor $72.173$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 450 = 2 \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.1727189158\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 30)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

\(f(q)\) \(=\) \( q + 4q^{2} + 16q^{4} - 32q^{7} + 64q^{8} + O(q^{10}) \) \( q + 4q^{2} + 16q^{4} - 32q^{7} + 64q^{8} - 12q^{11} + 154q^{13} - 128q^{14} + 256q^{16} - 918q^{17} - 1060q^{19} - 48q^{22} - 4224q^{23} + 616q^{26} - 512q^{28} + 7890q^{29} + 5192q^{31} + 1024q^{32} - 3672q^{34} - 16382q^{37} - 4240q^{38} - 3642q^{41} - 15116q^{43} - 192q^{44} - 16896q^{46} + 23592q^{47} - 15783q^{49} + 2464q^{52} - 16074q^{53} - 2048q^{56} + 31560q^{58} + 14340q^{59} - 47938q^{61} + 20768q^{62} + 4096q^{64} - 33092q^{67} - 14688q^{68} - 51912q^{71} - 12026q^{73} - 65528q^{74} - 16960q^{76} + 384q^{77} + 25160q^{79} - 14568q^{82} + 35796q^{83} - 60464q^{86} - 768q^{88} + 75510q^{89} - 4928q^{91} - 67584q^{92} + 94368q^{94} + 44158q^{97} - 63132q^{98} + O(q^{100}) \)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
4.00000 0 16.0000 0 0 −32.0000 64.0000 0 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 450.6.a.q 1
3.b odd 2 1 150.6.a.b 1
5.b even 2 1 90.6.a.a 1
5.c odd 4 2 450.6.c.i 2
15.d odd 2 1 30.6.a.b 1
15.e even 4 2 150.6.c.f 2
20.d odd 2 1 720.6.a.e 1
60.h even 2 1 240.6.a.f 1
120.i odd 2 1 960.6.a.d 1
120.m even 2 1 960.6.a.q 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.6.a.b 1 15.d odd 2 1
90.6.a.a 1 5.b even 2 1
150.6.a.b 1 3.b odd 2 1
150.6.c.f 2 15.e even 4 2
240.6.a.f 1 60.h even 2 1
450.6.a.q 1 1.a even 1 1 trivial
450.6.c.i 2 5.c odd 4 2
720.6.a.e 1 20.d odd 2 1
960.6.a.d 1 120.i odd 2 1
960.6.a.q 1 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(450))\):

\( T_{7} + 32 \)
\( T_{11} + 12 \)
\( T_{17} + 918 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( -4 + T \)
$3$ \( T \)
$5$ \( T \)
$7$ \( 32 + T \)
$11$ \( 12 + T \)
$13$ \( -154 + T \)
$17$ \( 918 + T \)
$19$ \( 1060 + T \)
$23$ \( 4224 + T \)
$29$ \( -7890 + T \)
$31$ \( -5192 + T \)
$37$ \( 16382 + T \)
$41$ \( 3642 + T \)
$43$ \( 15116 + T \)
$47$ \( -23592 + T \)
$53$ \( 16074 + T \)
$59$ \( -14340 + T \)
$61$ \( 47938 + T \)
$67$ \( 33092 + T \)
$71$ \( 51912 + T \)
$73$ \( 12026 + T \)
$79$ \( -25160 + T \)
$83$ \( -35796 + T \)
$89$ \( -75510 + T \)
$97$ \( -44158 + T \)
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